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桂林電子科技大學(xué)畢業(yè)設(shè)計(jì) 論文 報(bào)告用紙 摘 要 模具作為一種成型工具 其設(shè)計(jì) 制造水平的高低 直接關(guān)系到產(chǎn)品的質(zhì)量與更 新?lián)Q代 是衡量一個(gè)國(guó)家產(chǎn)品制造水平的重要標(biāo)志 支撐管彎頭體積較小 內(nèi)部結(jié)構(gòu)復(fù)雜 因而對(duì)注射成型模具和成型工藝的要求極 高 支撐管彎頭注射模設(shè)計(jì)制造的最大難點(diǎn)在于成型部件 澆注系統(tǒng) 脫模機(jī)構(gòu)的設(shè) 計(jì) 然后用 Solidworks 進(jìn)行三維實(shí)體建模 再進(jìn)行實(shí)體分析 確定出型腔數(shù)目 初選出成 型設(shè)備 確定塑件的擺放位置 然后進(jìn)行分型面的確定 澆口的確定 確定結(jié)構(gòu)草圖 再搭 配 Solidworks 的 moldflow 可以非常有效的進(jìn)行模架設(shè)計(jì) 然后進(jìn)行抽芯機(jī)構(gòu) 推出機(jī)構(gòu) 復(fù) 位機(jī)構(gòu)的設(shè)計(jì) 冷卻系統(tǒng)的設(shè)計(jì) 最后在 Solidworks 下進(jìn)行機(jī)構(gòu)模擬分析和校核 在這 過(guò)程中經(jīng)常發(fā)現(xiàn)不合理或者干涉的情況 然后分析這些狀況產(chǎn)生的原因 然后進(jìn)行修改 直到最后確定比較合理的方案 方案出來(lái)以后 再利用 Solidworks 的有限元分析功能對(duì) 模具的受力部分進(jìn)行強(qiáng)度校核 這也是 Solidworks 比較強(qiáng)大的功能之一 當(dāng)完全確定沒(méi) 有問(wèn)題的時(shí)候 就進(jìn)行二維圖形的繪制 在繪制二維圖形的時(shí)候 運(yùn)用目前機(jī)械行業(yè)最 有有效的二維圖紙繪制方法 由三維轉(zhuǎn)成二維 這不但在設(shè)計(jì)的過(guò)程思維更加清晰 把大量經(jīng)歷用于機(jī)構(gòu)的優(yōu)化和完善 最重要的是大大提高的繪圖速度和準(zhǔn)確性 這在目 前的經(jīng)濟(jì)時(shí)代是非常重要的 這套模具最重要的是通過(guò)傳統(tǒng)和現(xiàn)代二種思維方法來(lái)設(shè)計(jì)的 可以親身體會(huì)出二種 方法各自的優(yōu)缺點(diǎn) 取長(zhǎng)補(bǔ)短 可以讓傳統(tǒng)工業(yè)煥發(fā)出新的活力 也提高了大家學(xué)習(xí)興 趣 關(guān)鍵詞 注塑模具 支撐管彎頭 澆注系統(tǒng) 裝配工藝 桂林電子科技大學(xué)畢業(yè)設(shè)計(jì) 論文 報(bào)告用紙 Abstract Die as a tool for molding its design manufacture level are of direct bearing on the quality of products and replacement an important indicator to measure a country s level of manufacturing Support elbow is of small size and complex internal structure thus demanding a highly molding technique and injection die The most difficult parts of Support elbow injection mould design and manufacture are the design of molding part injection system stripping structures after demonstrated I choose low pressure polyethylene as the stuff Second I use Solidworks to do the three dimensional sculpting for the entity for the sake of deciding the number of swage equipment of injection and the place of produce Third I choose the parting line the gate the sketch of the machinery and arranged in pairs or groups imoldflow of Solidworks in this way we can design the mold s carrier with effectively Next the machine of take out fetch return and the cool system The end I use Solidworks to simulate the machine besides analyses and check it In the process I always find the phenomenon of inconsequence and interference when this phenomenon happened I must analyses what s wrong happened and why Until I fine the whys I must modify it The end I should decide the rational project After the project is putted forward I checked the intension of the pressed part of the mold with FEA of Solidworks FEA is one of the power functions of Solidworks After raveling out all problems I should draw the planar chart In the process of drawing transform the three dimensional chart to planar In this way my thinking became very in focus in the process of design so I spend mostly energy to optimize and consummate the machine the top drawer is that we can heighten speed and veracity of drawing At present this is very important In the process of the design I used traditionary technique and unconventional ways Compared them I found theirs strongpoint and disadvantage so I can learn from other s strong points to offset one s weakness Improving the traditionary technique besides improve our s interest to study machine Key Words Injection mold Support elbow Pouring system Assembly process 桂林電子科技大學(xué)畢業(yè)設(shè)計(jì) 論文 報(bào)告用紙 目 錄 1 緒 論 1 1 1 國(guó)內(nèi)外發(fā)展?fàn)顩r 1 1 1 1 模具工業(yè)的概況 1 1 1 2 我國(guó)塑料模具工業(yè)和技術(shù)狀況及地區(qū)分布 2 1 1 3 我國(guó)塑料模具工業(yè)和技術(shù)的今后的主要發(fā)展方向 5 1 1 4 注塑模具 CAD 發(fā)展概況及趨勢(shì) 5 1 2 研究?jī)?nèi)容 7 1 2 1 支撐管彎頭外形設(shè)計(jì) 7 1 2 2 分析最佳成型工藝 7 1 2 3 模具結(jié)構(gòu)分析和確定 7 1 2 4 模具開(kāi)合模運(yùn)動(dòng)仿真 7 2 支撐管彎頭設(shè)計(jì)及其成型工藝分析 8 2 1 制品結(jié)構(gòu)和形狀的設(shè)計(jì) 8 2 2 制品材料的選擇 9 2 2 1 丙烯腈 丁二烯 苯乙烯三元共聚物 ABS 9 2 2 2 聚苯乙烯 PS 9 2 2 3 雙酚 A 型碳酸脂 PC 10 2 3 注射工藝選擇 12 2 3 1 ABS 塑料的干燥 12 2 3 2 注射壓力 12 2 3 3 注射溫度 12 2 3 4 模具溫度 13 2 3 5 料量控制 13 3 模具設(shè)計(jì) 14 3 1 型腔數(shù)量的確定 14 3 2 注塑機(jī)選型 14 3 2 1 注射量計(jì)算 15 3 2 2 注射機(jī)型號(hào)確定 16 3 2 3 注射壓力校核 16 3 2 4 鎖模力校核 16 3 2 5 開(kāi)模行程和模板安裝尺寸校核 17 3 3 模具澆注系統(tǒng)設(shè)計(jì) 18 3 3 1 主流道設(shè)計(jì) 18 3 3 2 分流道 18 桂林電子科技大學(xué)畢業(yè)設(shè)計(jì) 論文 報(bào)告用紙 3 3 3 澆口設(shè)計(jì) 19 3 4 注射模具成型零部件設(shè)計(jì) 20 3 4 1 成型零部件尺寸分析 20 3 4 2 塑件收縮率的影響 20 3 4 3 成型零件的設(shè)計(jì) 21 3 4 3 1 型腔 21 3 4 3 2 側(cè)型芯 22 3 4 4 成型零部件強(qiáng)度校核計(jì)算 23 3 4 5 型芯與型腔配合 23 3 5 脫模機(jī)構(gòu)設(shè)計(jì) 24 3 6 側(cè)向抽芯設(shè)計(jì) 26 3 6 1 機(jī)構(gòu)設(shè)計(jì) 26 3 6 2 確定抽芯距 27 3 6 3 抽芯力的計(jì)算 27 3 6 4 斜導(dǎo)柱設(shè)計(jì) 28 3 7 導(dǎo)柱導(dǎo)向機(jī)構(gòu) 29 3 7 1 結(jié)構(gòu)形式 29 3 7 2 導(dǎo)柱結(jié)構(gòu)和技術(shù)要求 30 3 7 3 導(dǎo)套 30 3 8 模具溫度調(diào)節(jié)系統(tǒng) 31 3 9 模具材料 31 4 模具裝配工藝 33 4 1 塑料模具的裝配基準(zhǔn) 33 4 2 塑料模具的總裝配程序 33 4 3 塑料模具裝配時(shí)注意事項(xiàng) 34 4 4 空心球柄模具裝配工藝 34 結(jié) 語(yǔ) 36 致 謝 37 參考文獻(xiàn) 38 桂林電子科技大學(xué)畢業(yè)設(shè)計(jì) 論文 報(bào)告用紙 1 緒 論 1 1 國(guó)內(nèi)外發(fā)展?fàn)顩r 1 1 1 模具工業(yè)的概況 模具是機(jī)械 汽車 電子 通訊 家電等工業(yè)產(chǎn)品的基礎(chǔ)工藝裝備之一 作為工 業(yè)基礎(chǔ) 模具的質(zhì)量 精度 壽命對(duì)其他工業(yè)的發(fā)展起著十分重要的作用 在國(guó)際上 被稱為 工業(yè)之母 對(duì)國(guó)民經(jīng)濟(jì)發(fā)展起著不容質(zhì)疑的作用 模具工業(yè)是制造業(yè)中的一項(xiàng)基礎(chǔ)產(chǎn)業(yè) 是技術(shù)成果轉(zhuǎn)化的基礎(chǔ) 同時(shí)本身又是高 新技術(shù)產(chǎn)業(yè)的重要領(lǐng)域 在歐美等工業(yè)發(fā)達(dá)國(guó)家被稱為 點(diǎn)鐵成金 的 磁力工業(yè) 美國(guó)工業(yè)界認(rèn)為 模具工業(yè)是美國(guó)工業(yè)的基石 德國(guó)則認(rèn)為是所有工業(yè)中的 關(guān)鍵 工業(yè) 日本模具協(xié)會(huì)也認(rèn)為 模具是促進(jìn)社會(huì)繁榮富裕的動(dòng)力 同時(shí)也是 整個(gè) 工業(yè)發(fā)展的秘密 是 進(jìn)入富裕社會(huì)的原動(dòng)力 日本模具產(chǎn)業(yè)年產(chǎn)值達(dá)到13000億 日元 遠(yuǎn)遠(yuǎn)超過(guò)日本機(jī)床總產(chǎn)值9000億日元 如今 世界模具工業(yè)的發(fā)展甚至己超過(guò) 了新興的電子工業(yè) 在模具工業(yè)的總產(chǎn)值中 沖壓模具約占50 塑料模具約占33 壓鑄模具約占6 其它各類模具約占11 1 塑料模具工業(yè)是隨塑料工業(yè)的發(fā)展而發(fā)展的 塑料工業(yè)是一門新興工業(yè) 自塑料 問(wèn)世后的幾十年以來(lái) 由于其原料豐富 制作方便和成本低廉 塑料工業(yè)發(fā)展很快 它在某些方面己取代了多種有色金屬 黑色金屬 水泥 橡膠 皮革 陶瓷 木材和 玻璃等 成為各個(gè)工業(yè)部門不可缺少的材料 2 目前在國(guó)民經(jīng)濟(jì)的各個(gè)部門中都廣泛地使用著各式各樣的塑料制品 特別是在辦 公設(shè)備 照相機(jī) 汽車 儀器儀表 機(jī)械制造 交通 電信 輕工 建筑業(yè)產(chǎn)品 日 用品以及家用電器行業(yè)中的電視機(jī) 收錄機(jī) 洗衣機(jī) 電冰箱和手表的殼體等零件 都已經(jīng)向塑料化方向發(fā)展 近幾年來(lái)由于工程塑料制件的強(qiáng)度和精度等得到很大的提 高 因而各種工程塑料零件的使用范圍正在不斷擴(kuò)大 預(yù)計(jì)今后隨著微型電子計(jì)算機(jī) 的普及和汽車的微型化 塑料制件的使用范圍將會(huì)越來(lái)越大 塑料工業(yè)的生產(chǎn)量也將 迅速增長(zhǎng) 塑料的應(yīng)用將覆蓋國(guó)民經(jīng)濟(jì)所有部門 尤其在國(guó)防和尖端科學(xué)技術(shù)領(lǐng)域中 占有越來(lái)越重要的地位 目前 世界的塑料產(chǎn)量已超過(guò)有色金屬產(chǎn)量的總和 3 塑料模具就是利用特定形狀去成型具有一定形狀和尺寸的塑料制品的工藝基礎(chǔ)裝 備 用塑料模具生產(chǎn)的主要優(yōu)點(diǎn)是制造簡(jiǎn)便 材料利用高 生產(chǎn)率高 產(chǎn)品的尺寸規(guī) 格一致 特別是對(duì)大批量生產(chǎn)的機(jī)電產(chǎn)品 更能獲得價(jià)廉物美的經(jīng)濟(jì)效果 塑料模具 的現(xiàn)代設(shè)計(jì)與制造和現(xiàn)代塑料工業(yè)的發(fā)展有極密切的關(guān)系 隨著塑料工業(yè)的飛速發(fā)展 塑料模具工業(yè)也隨之迅速發(fā)展 Computer Aided Design 40 2008 space C L producti moulded part Despite the various research efforts that have been directed towards the analysis optimization and fabrication of cooling systems support for the layout design of the cooling system has not been well developed In the layout design phase a major concern is the feasibility of building the cooling system inside the mould insert without interfering with the other mould components This paper reports a configuration space C space method to address this important issue While a high dimensional C space is generally required to deal with a complex system such as a cooling system the special characteristics of cooling system design are exploited in the present study and special techniques that allow C space computation and storage in three dimensional or lower dimension are developed This new method is an improvement on the heuristic method developed previously by the authors because the C space representation enables an automatic layout design system to conduct a more systematic search among all of the feasible designs A simple genetic algorithm is implemented and integrated with the C space representation to automatically generate candidate layout designs Design examples generated by the genetic algorithm are given to demonstrate the feasibility of the method c 2007 Elsevier Ltd All rights reserved Keywords Cooling system design Plastic injection mould Configuration space method 1 Introduction The cooling system of an injection mould is very important to the productivity of the injection moulding process and the quality of the moulded part Extensive research has been conducted into the analysis of cooling systems 1 2 and commercial CAE systems such as MOLDFLOW 3 and Moldex3D 4 are widely used in the industry Research into techniques to optimize a given cooling system has also been reported 5 8 Recently methods to build better cooling systems by using new forms of fabrication technology have been reported Xu et al 9 reported the design and fabrication of conformal cooling channels that maintain a constant distance from the mould impression Sun et al 10 11 used CNC Despitethevariousresearcheffortsthathavefocusedmainly on the preliminary design phase of the cooling system design process in which the major concern is the performance of the cooling function of the system support for the layout design phase in which the feasibility and manufacturability of the cooling system design are addressed has not been well developed A major concern in the layout design phase is the feasibility of building the cooling system inside the mould insert without interfering with the other mould components Consider the example shown in Fig 1 It can be seen that many different components of the various subsystems of the injection mould such as ejector pins slides sub inserts and so forth have to be packed into the mould insert Finding the best location for each channel of the cooling circuit to optimize Plastic injection mould cooling configuration C G Li Department of Manufacturing Engineering and Engineering Received 3 May 2007 accepted Abstract The cooling system of an injection mould is very important to the milling to produce U shaped milled grooves for cooling channels and Yu 12 proposed a scaffolding structure for the design of conformal cooling Corresponding author E mail address meclli cityu edu hk C L Li 0010 4485 see front matter c 2007 Elsevier Ltd All rights reserved doi 10 1016 j cad 2007 11 010 334 349 system design by the method Li Management City University of Hong Kong Hong Kong 18 November 2007 vity of the injection moulding process and the quality of the the cooling performance of the cooling system and to avoid interference with the other components is not a simple task Another issue that further complicates the layout design problem is that the individual cooling channels need to be connected to form a path that connects between the inlet and the outlet Therefore changing the location of a channel may 335 Fig 1 Thecoolingsystem components require changing the example shown in to optimize the cooling in Fig 2 a Assume other mould components mould component As C1 cannot be mo interference with other C2 is moved and C connectivity as sho C3 is found to interfere mould components is very tedious that supports the this new technique used to provide a layout designs The an efficient method the layout design to generate layout system developed w C space method to conduct a more layout designs is the space that system is treated the configuration free region Points of the the components correspond to of the system initially formalized planning problems shortened and further modification is needed which results in the final design shown in Fig 2 c Given that a typical injection mould may have more than ten cooling channels with each channel a Interference occurs between cooling channel C1 and mould component O1 at the ideal location of C1 c C3 is moved and C2 is design Fig 2 An example showing the tediousness and a survey in this area of research has been reported by Wise and Bowyer 16 The C space method has also been used to solve problems in qualitative reasoning e g 17 18 b Channel C1 is shortened C2 is moved and C3 is elongated to give the final C G Li C L Li Computer Aided Design 40 2008 334 349 insideamouldinsertpackedwithmanyothermould other channels as well Consider the Fig 2 The ideal location of each channel performance of the system is shown that when the cooling system and the are built into the mould insert a O1 is found to interfere with channel C1 ved to a nearby location due to the possible components it is shortened As a result 3 is elongated accordingly to maintain the wn in Fig 2 b Owing to its new length with another mould component O2 potentially interfering with a few other finding an optimal layout design manually This paper reports a new technique automation of the layout design process In a configuration space C space method is concise representation of all of the feasible C space representation is constructed by that exploits the special characteristics of problem Instead of using heuristic rules designs as in the automatic layout design previously by the authors 13 14 this ne enables an automatic layout design system systematic search among all of the feasible 2 The configuration space method In general the C space of a system results when each degree of freedom of that as a dimension of the space Regions in space are labeled as blocked region or in the free regions correspond to valid configurations system where there is no interference between of the system Points in the blocked regions invalid configurations where the components interfere with one another C space was by Lozano Perez 15 to solve robot path of the layout design process 336 and e g automatic 23 2 1 the y c 3 se e a cooling system Fig 3 gives an example The preliminary design of this cooling system consists of four cooling channels To generate a layout design from the preliminary design the centers and lengths of the channels are adjusted As shown in Fig 3 the center of channel C1 can be moved along the X1 and X2 directions and its length can be adjusted along the X3 direction Similarly the length of C2 can be adjusted along the X4 direction while its center adjustment is described by X1 and X3 and thus must be the same as the adjustment of C1 to maintain the connectivity By applying similar arguments to the other channels it can be seen that the cooling system has 5 a Channel Ci and three mould components inside the mould insert b Offsets of the mould Ci represented by line d The initial free region of Ci Fig 4 The major steps in the construction considered To account for the diameter D Oi is first offset by D 2 M to give Oprimei where M is the minimum allowable distance between the channel wall and the face of a component This growing of Oi in effect reduces channel Ci to a line Li Consider the example illustrated in Fig 4 Fig 4 a shows a channel Ci and three mould components O1 O2 and O3 that may interfere with Ci Fig 4 b shows the offsets Oprime1 Oprime2 and Oprime3 of the mould components and the reduction of Ci to a line segment Li that is coincident with the axis of Ci If there is no intersection between Li and the offsets of the mould components then the original channel Ci will not intersect with components and gment Li c Sweeping the offsets of the mould components and Ci represented by point Pi Subtracting Oprimeprimei from Bprimei f The free region FRi of Ci C G Li C L Li Computer Aided Design 40 2008 334 349 Fig 3 An example showing the degrees of freedom of a cooling system the analysis and design automation of kinematic devices 19 21 TheauthorinvestigatedaC spacemethodinthe design synthesis of multiple state mechanisms 22 in previous research C space of a cooling system A high dimensional C space can be used to represent all of feasible layout designs of a given preliminary design of degreesoffreedom andtheyaredenotedas Xi i 1 2 5 In principle the C space is a five dimensional space and an point in the free region of this space gives a set of coordinate values on the Xi axes that can be used to define the geometry of the channels without causing interference with the other mould components Todeterminethefreeregioninahigh dimensional C spaceofacoolingsystem thefirststepistoconstructthefree regions in the C spaces of the individual channels 2 2 C space construction of individual cooling channels When an individual channel Ci is considered alone it has three degrees of freedom say X1 and X2 for its center location and X3 for its length As the ideal center location and length have already been specified in the preliminary design it is reasonable to assume a fixed maximum allowable variation for X1 X2 and X3 The initial free region in the C space of channel Ci is thus a three dimensional cube Bi with the dimensions c c c To avoid any possible interference with a mould component Oi when channel Ci is built into the mould insert by drilling a drilling diameter D and drilling depth along X have to be of the free region FRi of a channel Ci C G Li C L Li Computer Aided the mould components This growing or offset of an obstacle is a standard technique in the C space method 15 A channel is formed by drilling from a face of the mould insert and any obstacle Oi within the drilling depth will affect the construction of the channel To account for the drilling depth the offset Oprimei of Oi is swept along the drilling direction until the opposite face of the mould insert is reached to generate Oprimeprimei This sweeping of Oprimei in effect reduces line Li to a point Pi located at the end of Li As shown in Fig 4 c if the point Pi is outside Oprimeprimei the drilling along Li to produce Ci is feasible The free region FRi of channel Ci is obtained as follows First the initial free region Bi is constructed with its center at Pi as shown in Fig 4 d Bi then intersects with the mould insert to obtain Bprimei Bprimei represents all of the possible variations of Ci when only the geometric shape of the mould insert is considered Then FRi is obtained by subtracting from Bprimei the Oprimeprimei of all of the obstacles Fig 4 e and f show the subtraction and the resulting FRi of the example 2 3 Basic approach to the construction of the C space of cooling system To determine the free region FRF in the C space of a cooling system the free regions of each cooling channel have to be intersected in a proper manner so that the effect of the obstacles to all of the channels are properly represented by FRF However the standard Boolean intersection between the free regions of two different channels cannot be performed because their C spaces are in general spanned by different sets of axes Referring to the example in Fig 3 the C spaces of C1 and C2 are spanned by X1 X2 X3 and X1 X3 X4 respectively To facilitate the intersection between free regions in different C spaces the projection of a region from the C space of one channel to that of another channel is needed The following notations are first introduced and will be used in the subsequent discussions on projections and the rest of the paper Notations used in describing high dimensional spaces Sn denotes an n dimensional space spanned by the set of axes Xn X1 X2 Xn Sm denotes an m dimensional space spanned by the set of axes Xm Xprime1 Xprime2 Xprimem pn denotes a point in Sn and pn x1 x2 xn where xi denotes a coordinate on the ith axis Xi Rn denotes a region in Sn Rn Sn Rn is a set of points in Sn PROJSm pn denotes the projection of a point pn from Sn to Sm PROJSm Rn denotes the projection of a region Rn from Sn to Sm Notations used in describing a cooling system nC denotes the number of channels in the cooling system nF denotes the total degrees of freedom of the cooling system Ci denotes the ith channel of the cooling system Si denotes the C space of Ci Design 40 2008 334 349 337 FRi denotes the free region in Si That is it is the free region of an individual channel Ci SF denotes the C space of the cooling system FRF denotes the free region in SF That is it is the free region of the cooling system Consider the projection of a point pn in Sn to a point pm in Sm Fig 5 a illustrates examples of projection using spaces of one dimension to three dimensions Projections are illustrated forthreecases i Xm Xn ii Xm Xn and iii Xm negationslash Xn Xn negationslash Xm and Xn Xm negationslash For i each coordinate of pm is equal to a corresponding coordinate of pn that is on the same axis For ii and iii the projection of pn is a region Rm For each point pm in Rm a coordinate of pm is equal to that of pn if that coordinate is on a common axis of Sn and Sm For the other coordinates of pm any value can be assigned The reason for this specific definition of the projections in particular for cases ii and iii is as follows Consider two adjacent channels Cn and Cm As they are adjacent they must be connected and thus their C spacesSn and Sm share some common axes Assume that a configuration that corresponds to a point pn in Sn has been selected for Cn To maintain the connectivity the configuration for Cm must be selected such that the corresponding point pm in Sm shares the same coordinates with pn on their common axes This implies that pm can be any point within the projection of pn on Sm where the method of projection is defined above The projections of a region Rn in Sn to Sm are simply the projections of every point in Rn to Sm Fig 5 b illustrates the region projections The formal definition of projection is given below Definition 1 Projection 1 1 If Xm Xn PROJSm pn is a point pm xprime1 xprime2 xprimem where for Xprimei X j xprimei xj for all i 1 m To simplify the notations in subsequent discussion this projection is regarded as a region that consists of the single point pm That is PROJSm pn pm 1 2 If Xm Xn PROJSm pn is a region Rm pm PROJSn pm pn 1 3 If Xm negationslash Xn Xn negationslash Xm and Xn Xm negationslash PROJSm pn is a region Rm pm PROJSI pm PROJSI pn where SI is the space spanned by Xn Xm If Xn Xm PROJSm pn is defined as Sm 1 4 PROJSm Rn is defined as the region Rm pm pm PROJSm pn pn Rn As discussed in Section 2 1 any point pF in FRF gives a value for each degree of freedom of the cooling system so that the geometry of the channels is free from interference with the other mould components In other words the projection of pF to each Si is in the free region FRi of each Ci Thus FRF is defined as follows Definition 2 Free Region in the C space of a Cooling System FRF pF PROJSi pF FRi i 1 nC Aided Note that according to to Si always contains only that span Si is always a subset The construction of the already been explained in the following theorem is useful Theorem 1 FRF nCintersectiondisplay i 1 PROJSF FRi Intuitively this theorem says first projected to the C space can then be obtained by performing among the projections The used in the proof are given of the C space F and to facilitate the between the regions can use a kind of cell used in 21 24 The region RF in Each box is defined by SF The intersection of of the two sets of high dimensional boxes intervals of each of the by m three OJSF FRi can then be boxes The construction Fig 5 The projections of points and regions in Sn to Sm Definition 1 1 the projection of pF a single point because the set of axes of the axes that span Sn free region FRi of each Ci has Section 2 2 To find FRF from FRi that to find FRF all of the FRi are of the cooling system SF FRF the Boolean intersections proof of Theorem 1 and the lemmas 2 4 Representation and computation To represent the free region FR computation of the Boolean intersections in a high dimensional space we enumeration method similar to the one basic idea is to approximate a high dimensional SF by a set of high dimensional boxes specifying an interval on each axis of two regions is achieved by the intersection boxes The intersection between two is simply the intersection between the boxes in each axis Assuming that each FRi is approximated dimensional boxes the projection PR approximated by mnF dimensional 338 C G Li C L Li Computer in the Appendix Design 40 2008 334 349 of FRF that uses Theorem 1 then requires mnC intersections between nF dimensional maximum of mnCnF of boxes used to represent intersections and FR is anticipated that the are still major problems improved method is 3 An efficient technique To avoid the high for the representation Instead we process to example shown in is assumed in this along the Z direction hasfourdegrees each channel Ci are shown in Fig 6 b channel C1 First a a A simple cooling system with four channels and four degrees of freedom b The free region FRi of each channel in its configuration space Si Fig 6 A simplified example of a cooling system design boxes and FRF is represented by a dimensional boxes Although the number the intermediate results of the F can be reduced by special techniques it memory and computational requirements of this method In the next section an developed for C space construction to represent and not to compute FRF explicitly focus on a technique that enables the computational work on the C spaces of each individual channel First consider the simplified design Fig 6 For the purpose of illustration it example that there is no variation in FRi ofthemouldinsertandthusthecoolingsystem of freedom as shown in Fig 6 a The Si of two dimensional and the assumed FRi are Consider a simple method for designing C G Li C L Li Computer Aided memory and computational requirements and construction of FRF we choose not Design 40 2008 334 349 339 point p1 can be selected from within FR1 so that C1 is free from interference with any obstacle However S1 is spanned Aided continued even though their C spaces of C1 i e they are as well because the system are connected have an effect in the cooling system To develop a design of each individual channels selection of a point always exist a corresponding that all of the channels system To address this Si is needed Definition 3 PRi is PRi PROJSi FRF Obviously for an always a correspondi FR2 Again as p2 x3 must have a value FR3 Also as must also be inside p1 p2 p3 and p4 C1 determine the valid designs for C1 the The effect of FR4 valid region in FR3 finally in S1 The all of the effects of is formally channels Ci and of their free regions do not have an axis common to that not adjacent to C1 have to be considered cooling channels that make up the cooling A choice in one degree of freedom will choice of another degree of freedom of the process that works on the C spaces a major concern is that after the in the C space of one channel there must point in all of the other Si such can be connected to form a valid cooling concern the projection of FRF to each defined as the projection of FRF to Si which we can find a p2 x2 x3 within has a coordinate x3 in X3 the coordinate for which